Previous microphones have been developed primarily for use in sound reinforcement systems and for monophonic and stereophonic recording. Pressure microphones have an omnidirectional response, being equally sensitive to sounds arriving from all directions. First order gradient microphones were developed to provide a variety of directional responses, which can increase the potential acoustic gain in sound reinforcement systems in reverberant environments. These microphones also allow stereophonic recording with acceptable imaging within the loudspeaker angles. The gradient microphone is in many cases implemented as two closely spaced pressure elements with their outputs subtracted. This produces an approximation to the gradient, and a signal proportional to the sound velocity is obtained by integrating the difference signal.
Second order gradient microphones have also been developed which provide greater discrimination between sound from different angles of arrival. These typically consist of two gradient elements—each often consisting of two pressure elements—which produce the second spatial derivative with respect to one, or two axes. A pure second order response is obtained using the derivative with respect to two axes, and the four pressure elements form a square with their outputs combined with amplitudes of plus or minus one. This array produces a sin (2θ) polar response. A second square array is obtained by rotating the first by 45 producing a cos (2θ) response. If the outputs are integrated twice, then at low frequencies the response is constant with frequency. Alternative implementations consist of two pressure gradient elements, or a single diaphragm open to the atmosphere at four points, with two openings to one side of the diaphragm and two openings connected to the other to produce the appropriate signs.
Higher order devices may also be built using three or more gradient elements and similar implementation methods to that of the second order microphones. For each order m, an mth order integration is required to produce a flat response with frequency.
An alternative method for improving the discrimination of a microphone is to use two or more individual microphones, and to combine their outputs to produce one or more outputs which have higher directivity than a single element. More complex systems may be built using a larger array of microphones. Typically, prior art examples consist of a straight line of microphones with either equal or different inter-microphone separations, and use beam forming principles to produce one or more beams with sharp directivity in one or more directions.
Surround sound systems offer the potential for improved sound localisation over stereo systems. Early quadraphonic systems brought to light some of the issues that affect the quality of reproduction, in particular the limitations of small numbers of loudspeakers, and the importance of the functions used to place individual sound sources in the 360 degree sound field. The ambisonics system was developed independently by several researchers, and has proved to be a low order approximation to the holographic reconstruction of sound fields. The sound field is recorded using microphones that measure the spherical harmonics of the sound field at (theoretically) a point. The performance of the system becomes more accurate over wider areas as the number of loudspeakers and the number of spherical harmonics of the recorded sound field are increased.
All current ambisonics systems are first order: that is, they use a recording microphone which records only the zeroth (pressure) and first (x, y and z components of velocity) responses. A prior art microphone designed specifically for this purpose is the Soundfield microphone. Since only the first spherical harmonic, also termed spatial harmonic in the art, is available, the resulting reproduction demonstrates poor localisation.
Most surround systems use only the horizontal (x and y) components of the velocity, since a) lateral localisation is more acute than vertical localisation, and b) the use of the z component requires loudspeakers to be positioned above the listener, which is often impractical. In this case the spatial harmonics are obtained from microphones with azimuthal polar responses of the form cos (mθ) and sin (mθ). Each spherical harmonic greater than order zero therefore requires 2 channels. The total number of channels required to transmit or record all spatial harmonics up to order M is thus 2M+1.
Modem surround sound systems typically use five loudspeakers, and it has been shown that this allows the use of microphones which can measure up to the second order spherical harmonics of the sound field, requiring five channels. Surround systems using more than five loudspeakers will allow harmonics of orders greater than 2, and higher numbers of channels are required—for example, the inclusion of third order spherical harmonics require seven channels.
The recently introduced DVD-Audio disk allows the recording of six channels of audio. It is thus capable of carrying recordings from second order microphone systems. Future audio disk technology will provide greater numbers of channels. While some second and higher order microphones have been developed in the past, there are currently no microphone systems commercially available which can measure spherical harmonics of order two or greater. There is thus a technology mismatch between the reproduction capability that DVD disks offer and the recording technology that current microphones can provide. A practical need therefore exists for the development of microphone systems that can accurately record the higher spherical harmonics of sound fields in the horizontal plane, and in particular, the second order responses.
Consider a general sound pressure field p(x,y,z,t). The pressure in the plane z=0 is a three-dimensional function of x,y and t. This three-dimensional function may be equivalently expressed in terms of its three-dimensional Fourier transform
                              P          ⁡                      (                                          k                ->                            ,              ω                        )                          =                  ∫                                    ∫                              -                ∞                            ∞                        ⁢                                          p                ⁡                                  (                                      x                    ,                    y                    ,                    t                                    )                                            ⁢                              ⅇ                                  -                                      j                    ⁡                                          [                                                                        ω                          ⁢                                                                                                          ⁢                          t                                                +                                                                              k                            ->                                                    ·                                                      r                            ->                                                                                              ]                                                                                  ⁢                                                          ⁢                              ⅆ                t                            ⁢                              ⅆ                                  r                  ->                                                                                        (        1        )            where {right arrow over (k)} is the vector wavenumber and (−j{right arrow over (k)}·{right arrow over (r)}) is chosen so that the pressure is represented by incoming waves which is relevant in surround systems, as opposed to outgoing waves in some texts. This equations shows that any sound field in the horizontal plane z=0 can be expressed as a sum of plane waves.
Writing {right arrow over (k)} in terms of its two components u=k cos (θ) and ν=k sin (θ), where k=|{right arrow over (k)}|, this may be written
                              P          ⁡                      (                          u              ,              v              ,              ω                        )                          =                              ∫                          -              ∞                        ∞                    ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                p                  ⁡                                      (                                          x                      ,                      y                      ,                      t                                        )                                                  ⁢                                  ⅇ                                      -                                          j                      ⁡                                              (                                                                              ω                            ⁢                                                                                                                  ⁢                            t                                                    +                          ux                          +                          vy                                                )                                                                                            ⁢                                                                  ⁢                                  ⅆ                  t                                ⁢                                                                  ⁢                                  ⅆ                  x                                ⁢                                                                  ⁢                                  ⅆ                  y                                                                                        (        2        )            
As an example, a complex plane wave with radian frequency ω0, magnitude B, phase φ and angle of incidence θ0 has the formp(x, y, t)=Bej[ω0t+φ+k0 cos (θu)x+k0 sin (θ0)y]  (3)where k0=ω0/c and c is the speed of sound. The Fourier transform isP(u, v, ω)=A(2π)3 δ(u−k0 cos (θ0))δ(v−k0 sin (θ0))δ(ω−ω0)   (4)where, for convenience, A=BeJφ is the complex amplitude. The “spectrum” consists of a delta function at as ω=ω0, u=k cos (θ0), ν=k sin (θ0). Since P(u, v, ω) exists only at one point, it may be represented as a vector 10 in wavenumber-frequency space 11, as shown in FIG. 1. In the (u,v) plane, the vector 10 has a projection 12 which is a vector of radius k0=ω0/c and angle θ0 relative to the u axis.
A real plane wave is given by the real part of equation 3,
                                          p            R                    ⁡                      (                          x              ,              y              ,              t                        )                          =                                            1              2                        ⁢            A            ⁢                                                  ⁢                          ⅇ                              j                ⁡                                  [                                                                                    ω                        0                                            ⁢                      t                                        +                                                                  k                        0                                            ⁢                                              cos                        ⁡                                                  (                                                      θ                            0                                                    )                                                                    ⁢                      x                                        +                                                                  k                        0                                            ⁢                                              sin                        ⁡                                                  (                                                      θ                            0                                                    )                                                                    ⁢                      y                                                        ]                                                              +                                    1              2                        ⁢                                          A                ⁢                                                                              *                        ⁢                          ⅇ                              -                                  j                  ⁡                                      [                                                                                            ω                          0                                                ⁢                        t                                            +                                                                        k                          0                                                ⁢                                                  cos                          ⁡                                                      (                                                          θ                              0                                                        )                                                                          ⁢                        x                                            +                                                                        k                          0                                                ⁢                                                  sin                          ⁡                                                      (                                                          θ                              0                                                        )                                                                          ⁢                        y                                                              ]                                                                                                          (        5        )            which can be written
                                          p            R                    ⁡                      (                          x              ,              y              ,              t                        )                          =                                            1              2                        ⁢            A            ⁢                                                  ⁢                          ⅇ                                                                    jω                    0                                    ⁢                  t                                +                                                      jk                    0                                    ⁡                                      [                                                                                            cos                          ⁡                                                      (                                                          θ                              0                                                        )                                                                          ⁢                        x                                            +                                                                        sin                          ⁡                                                      (                                                          θ                              0                                                        )                                                                          ⁢                        y                                                              ]                                                                                +                                    1              2                        ⁢                                          A                ⁢                                                                              *                        ⁢                          ⅇ                                                                    -                                          jω                      0                                                        ⁢                  t                                +                                                      k                    0                                    ⁡                                      [                                                                                            cos                          ⁡                                                      (                                                                                          θ                                0                                                            +                              x                                                        )                                                                          ⁢                        x                                            +                                                                        k                          0                                                ⁢                                                  sin                          ⁡                                                      (                                                                                          θ                                0                                                            +                              π                                                        )                                                                          ⁢                        y                                                              ]                                                                                                          (        6        )            
The second term consists of a negative frequency complex plane wave with conjugate phase and the same positive wavenumber k0 propagating in the opposite direction θ0+π. The spectrum may be represented as two vectors in (u, v, ω) space. As ω0 and θ0 vary, the two vectors trace out a cone shape, since k=ω/c. Thus the spectrum of any two-dimensional spatial pressure field lies in the cone ω=±ck in the three-dimensional (u, v, ω) space.
The pressure field is obtained from P(u, v, ω) by the inverse Fourier transform
                              p          ⁡                      (                          x              ,              y              ,              t                        )                          =                              1                                          (                                  2                  ⁢                  π                                )                            3                                ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      P                    ⁡                                          (                                              u                        ,                        v                        ,                        ω                                            )                                                        ⁢                                      ⅇ                                          j                      ⁡                                              [                                                                              ω                            ⁢                                                                                                                  ⁢                            t                                                    +                          uv                          +                          vy                                                ]                                                                              ⁢                                                                          ⁢                                      ⅆ                    u                                    ⁢                                                                          ⁢                                      ⅆ                    v                                    ⁢                                                                          ⁢                                      ⅆ                    ω                                                                                                          (        7        )            
Writing P(u, v, ω) in terms of spatial polar coordinates, u=k cos (θ), v=k sin (θ), and p(x, y, t) in terms of polar coordinates x=r cos (θ), y=r sin (θ) yields
                              p          ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              1                                          (                                  2                  ⁢                  π                                )                            3                                ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                0                ∞                            ⁢                                                ∫                  0                                      2                    ⁢                    π                                                  ⁢                                                      P                    ⁡                                          (                                              k                        ,                        θ                        ,                        ω                                            )                                                        ⁢                                      ⅇ                                          j                      ⁡                                              [                                                                              ω                            ⁢                                                                                                                  ⁢                            t                                                    +                                                      kr                            ⁢                                                                                                                  ⁢                                                          cos                              ⁡                                                              (                                                                  θ                                  -                                  ϕ                                                                )                                                                                                                                    ]                                                                              ⁢                  k                  ⁢                                                                          ⁢                                      ⅆ                    k                                    ⁢                                                                          ⁢                                      ⅆ                    θ                                    ⁢                                                                          ⁢                                      ⅆ                    ω                                                                                                          (        8        )            
Since k=ω/c the integral over ω is only nonzero for ω=±kc. HenceP(k, θ, ω)=P(k, θ, ω)2π[δ(ω−kc)+δ(ω+kc)]  (9)and so
                              p          ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                                            1                              4                ⁢                                  π                  2                                                      ⁢                                          ∫                0                ∞                            ⁢                                                ∫                  0                                      2                    ⁢                    π                                                  ⁢                                                      P                    ⁡                                          (                                              k                        ,                        θ                        ,                        kc                                            )                                                        ⁢                                      ⅇ                                          j                      ⁢                                                                                          ⁢                                              k                        ⁡                                                  [                                                      ct                            +                                                          r                              ⁢                                                                                                                          ⁢                                                              cos                                ⁡                                                                  (                                                                      θ                                    -                                    ϕ                                                                    )                                                                                                                                              ]                                                                                                      ⁢                                                                          ⁢                  k                  ⁢                                      ⅆ                    k                                    ⁢                                                                          ⁢                                      ⅆ                    θ                                                                                +                                    1                              4                ⁢                                                                  ⁢                                  π                  2                                                      ⁢                                          ∫                0                ∞                            ⁢                                                ∫                  0                                      2                    ⁢                    π                                                  ⁢                                                      P                    ⁡                                          (                                              k                        ,                        θ                        ,                                                  -                          kc                                                                    )                                                        ⁢                                      ⅇ                                          j                      ⁢                                                                                          ⁢                                              k                        ⁡                                                  [                                                                                    -                              ct                                                        +                                                          r                              ⁢                                                                                                                          ⁢                                                              cos                                ⁡                                                                  (                                  θ–ϕ                                  )                                                                                                                                              ]                                                                                                      ⁢                  k                  ⁢                                                                          ⁢                                      ⅆ                    k                                    ⁢                                                                          ⁢                                      ⅆ                    θ                                                                                                          (        10        )            
There are two special cases of interest. In the first, the signal contains only positive frequencies, (for example the complex plane wave considered above) and the pressure field is analytic. In this case the second integral is zero, and the analytic pressure field is
                                          p            a                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              1                          4              ⁢                              π                2                                              ⁢                                    ∫              0              ∞                        ⁢                                          ∫                0                                  2                  ⁢                  π                                            ⁢                                                P                  ⁡                                      (                                          k                      ,                      θ                      ,                      kc                                        )                                                  ⁢                                  ⅇ                                      j                    ⁢                                                                                  ⁢                                          k                      ⁡                                              [                                                  ct                          +                                                      r                            ⁢                                                                                                                  ⁢                                                          cos                              ⁡                                                              (                                                                  θ                                  -                                  ϕ                                                                )                                                                                                                                    ]                                                                                            ⁢                                                                  ⁢                k                ⁢                                  ⅆ                  k                                ⁢                                                                  ⁢                                  ⅆ                  θ                                                                                        (        11        )            
The analytic case is useful for the analysis and design of surround systems.
The second case of interest is real pressure fields, which occur in practice. In this case the spectrum in polar coordinates has the propertyP(k, θ, −kc)=P*(k, θ+π, kc)  (12)
Substituting this in equation 10
                                          p            R                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              1                          4              ⁢                              π                2                                              ⁢                                    ∫              0              ∞                        ⁢                                          ∫                0                                  2                  ⁢                  π                                            ⁢                              Re                ⁢                                  {                                                            P                      ⁡                                              (                                                  k                          ,                          θ                          ,                          kc                                                )                                                              ⁢                                          ⅇ                                              j                        ⁢                                                                                                  ⁢                                                  k                          ⁡                                                      [                                                          ct                              +                                                              r                                ⁢                                                                                                                                  ⁢                                                                  cos                                  ⁡                                                                      (                                                                          θ                                      -                                      ϕ                                                                        )                                                                                                                                                        ]                                                                                                                                }                                ⁢                                                                  ⁢                k                ⁢                                  ⅆ                  k                                ⁢                                                                  ⁢                                  ⅆ                  θ                                                                                        (        13        )            
Equations 11 and 13 both show that the pressure field is completely specified by a two dimensional spectrum S(k, θ)=kP(k, θ, kc) which specifies at each frequency, the complex amplitude of the plane wave arriving from each angle θ. S(k, θ) may be termed the frequency-dependent source distribution. Since it is periodic in θ, it can be expanded in a Fourier series
                              S          ⁡                      (                          k              ,              θ                        )                          =                  ∑                                                    q                m                            ⁡                              (                k                )                                      ⁢                          ⅇ                              j                ⁢                                                                  ⁢                m                ⁢                                                                  ⁢                θ                                                                        (        14        )            
The coefficients qm(k) are thus the “angular spectrum” of S(k, θ) at each spatial frequency k, given by
                                          q            m                    ⁡                      (            k            )                          =                              1                          2              ⁢              π                                ⁢                                    ∫              0                              2                ⁢                π                                      ⁢                                          S                ⁡                                  (                                      k                    ⁢                                                                                  ,                    θ                                    )                                            ⁢                              ⅇ                                                      -                    j                                    ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                  θ                                            ⁢                                                          ⁢                              ⅆ                θ                                                                        (        15        )            
The analysis is further simplified by examining each frequency component separately. In this case the sound field is “monochromatic”, consisting of complex plane waves of the same frequency ω0 arriving from all directions θ. In this case
                              P          ⁡                      (                          k              ,              θ              ,              kc                        )                          =                                            1              k                        ⁢                          S              ⁡                              (                                  k                  ,                  θ                                )                                              =                                                    2                ⁢                π                                            k                0                                      ⁢                          δ              ⁡                              (                                  k                  -                                      k                    0                                                  )                                      ⁢                                          S                0                            ⁡                              (                θ                )                                                                        (        16        )            where S0(θ)=S(k0, θ). Substituting this in equation 11 yields
                                          p            0                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              ⅇ                          j              ⁢                                                          ⁢                              ω                0                            ⁢              t                                ⁢                      1                          2              ⁢              π                                ⁢                                    ∫              0                              2                ⁢                π                                      ⁢                                                            S                  0                                ⁡                                  (                  θ                  )                                            ⁢                              ⅇ                                  j                  ⁢                                                                          ⁢                  kr                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                              θ                        -                        ϕ                                            )                                                                                  ⁢                                                          ⁢                              ⅆ                θ                                                                        (        17        )            
Thus a monochromatic sound field is expressed in terms of its one-dimensional source distribution. A simple example is a single plane wave with complex amplitude Λ arriving from direction θ0. The source distribution is a delta function at θ=θ0 and thus
                                          S                          0              ⁢                                                          ⁢                              θ                0                                              ⁡                      (            θ            )                          =                              2            ⁢            π            ⁢                                                  ⁢            A            ⁢                                          ∑                                  m                  =                                      -                    ∞                                                  ∞                            ⁢                              δ                ⁡                                  (                                      θ                    -                                          θ                      0                                        -                                          2                      ⁢                      m                      ⁢                                                                                          ⁢                      π                                                        )                                                              =                      A            ⁢                                          ∑                                  m                  =                                      -                    ∞                                                  ∞                            ⁢                              ⅇ                                  j                  ⁢                                                                          ⁢                                      m                    ⁡                                          (                                              θ                        -                                                  θ                          0                                                                    )                                                                                                                              (        18        )            and so the angular spectrum isqm=Ae−jmθ0   (19)
The monochromatic sound field may be written directly in terms of the spectrum of S0(θ) by substituting from equation 14,
                                          p            0                    ⁡                      (                          r              ,                                                          ⁢              ϕ              ,              i                        )                          =                              ⅇ                          j              ⁢                                                          ⁢                              ω                0                            ⁢              t                                ⁢                                    ∑                              m                =                                  -                  ∞                                            ∞                        ⁢                                          q                m                            ⁢                              1                                  2                  ⁢                  π                                            ⁢                                                ∫                  0                                      2                    ⁢                    π                                                  ⁢                                                      ⅇ                                          j                      ⁡                                              [                                                                              m                            ⁢                                                                                                                  ⁢                            θ                                                    +                                                                                    k                              0                                                        ⁢                            r                            ⁢                                                                                                                  ⁢                                                          cos                              ⁡                                                              (                                                                  θ                                  -                                  ϕ                                                                )                                                                                                                                    ]                                                                              ⁢                                                                          ⁢                                      ⅆ                    θ                                                                                                          (        20        )            which, with the identity
                                          j            n                    ⁢                                    J              n                        ⁡                          (              z              )                                      =                              1                          2              ⁢              π                                ⁢                                    ∫              0                              2                ⁢                π                                      ⁢                                          ⅇ                                  j                  ⁡                                      [                                                                  n                        ⁢                                                                                                  ⁢                        θ                                            +                                              z                        ⁢                                                                                                  ⁢                                                  cos                          ⁡                                                      (                            θ                            )                                                                                                                ]                                                              ⁢                                                          ⁢                              ⅆ                θ                                                                        (        21        )            yields
                                          p            0                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              ⅇ                          j              ⁢                                                          ⁢                              ω                0                            ⁢              t                                ⁢                                    ∑                              m                =                                  -                  ∞                                            ∞                        ⁢                                          j                m                            ⁢                                                J                  m                                ⁡                                  (                                                            k                      0                                        ⁢                    r                                    )                                            ⁢                              q                m                            ⁢                              ⅇ                                  j                  ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                  ϕ                                                                                        (        22        )            
This shows that the angular pressure field at radius r may be written as a sum of terms of the form exp(jmφ). These have been termed “phase modes” in antenna array literature and the same terminology will be used here. The magnitude of each phase mode is the spectral coefficient multiplied by a Bessel function of the first kind which describes how the phase mode varies radially.
An important feature of equation 22 is that for small k0r the Bessel functions of high orders are small and may be neglected without significantly affecting the pressure. Hence, for low frequencies, or for small radii, the phase mode expansion may be truncated to some maximum order m=±M . However, as the frequency or radius increases, M must increase to preserve the accuracy of the expression.
As an example, the pressure due to a single plane wave at angle θ0 is obtained from equations 19 and 22 with qm=A exp (−jmθ0)
                                          p                          0              ⁢                              θ                0                                              ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                  A          ⁢                                          ⁢                      ⅇ                          j              ⁢                                                          ⁢                              ω                0                            ⁢              t                                ⁢                                    ∑                              m                =                                  -                  ∞                                            ∞                        ⁢                                          j                m                            ⁢                                                J                  m                                ⁡                                  (                                                            k                      0                                        ⁢                    r                                    )                                            ⁢                              ⅇ                                  j                  ⁢                                                                          ⁢                                      m                    ⁡                                          (                                              ϕ                        -                                                  θ                          0                                                                    )                                                                                                                              (        23        )            
Thus the pressure field due to a plane wave consists of phase modes with magnitudes given by Bessel functions.
By adding the terms in equation 22 m=l and m=−l, and noting that J−m(z)=(−1)m Jm(z), the phase mode expansion may be written
                                                        p              0                        ⁡                          (                              r                ,                ϕ                ,                t                            )                                =                                    ⅇ                              j                ⁢                                                                  ⁢                                  ω                  0                                ⁢                t                                      ⁢                          [                                                q                  0                                ⁢                                                      J                    0                                    ⁡                                      (                    kr                    )                                                              +                                                ∑                                      m                    =                    1                                    ∞                                ⁢                                                                            j                      m                                        ⁡                                          [                                                                        q                          m                                                +                                                  q                                                      -                            m                                                                                              ]                                                        ⁢                                                            J                      m                                        ⁡                                          (                                                                        k                          0                                                ⁢                        r                                            )                                                        ⁢                                      cos                    ⁡                                          (                                              m                        ⁢                                                                                                  ⁢                        ϕ                                            )                                                                                  +                              j                ⁢                                                      ∑                                          m                      =                      1                                        ∞                                    ⁢                                                                                    j                        m                                            ⁡                                              [                                                                              q                            m                                                    -                                                      q                                                          -                              m                                                                                                      ]                                                              ⁢                                                                  J                        m                                            ⁡                                              (                                                                              k                            0                                                    ⁢                          r                                                )                                                              ⁢                                          sin                      ⁡                                              (                                                  m                          ⁢                                                                                                          ⁢                          ϕ                                                )                                                                                                                                ]                            (        24        )            
Thus the pressure may be alternatively written as a sum of cosine and sine terms, which are known as amplitude modes. In cases where the spectrum of S(θ) is Hermitian (q−m=qmm), this can be written
                                          p            0                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                              ⅇ                          j              ⁢                                                          ⁢                              ω                0                            ⁢              t                                ⁡                      [                                                            q                  0                                ⁢                                                      J                    0                                    ⁡                                      (                    kr                    )                                                              +                              2                ⁢                                                      ∑                                          m                      =                      1                                        ∞                                    ⁢                                                            j                      m                                        ⁢                                                                                  ⁢                    Re                    ⁢                                          {                                              q                        m                                            }                                        ⁢                                                                  J                        m                                            ⁡                                              (                                                                              k                            0                                                    ⁢                          r                                                )                                                              ⁢                                          cos                      ⁡                                              (                                                  m                          ⁢                                                                                                          ⁢                          ϕ                                                )                                                                                                        -                              2                ⁢                                                      ∑                                          m                      =                      1                                        ∞                                    ⁢                                                            j                      m                                        ⁢                    Im                    ⁢                                          {                                              q                        m                                            }                                        ⁢                                                                  J                        m                                            ⁡                                              (                                                                              k                            0                                                    ⁢                          r                                                )                                                              ⁢                                          sin                      ⁡                                              (                                                  m                          ⁢                                                                                                          ⁢                          ϕ                                                )                                                                                                                  ]                                              (        25        )            
The spectrum of the plane wave (equation 19) is Hermitian, and substituting for qm yields the simpler and well-known form
                                          p            0                    ⁡                      (                          r              ,              ϕ              ,              t                        )                          =                  A          ⁢                                          ⁢                                    ⅇ                                                jω                  0                                ⁢                t                                      ⁡                          [                                                                    J                    0                                    ⁡                                      (                    kr                    )                                                  +                                  2                  ⁢                                                            ∑                                              m                        =                        1                                            ∞                                        ⁢                                                                                  ⁢                                                                  j                        m                                            ⁢                                                                        J                          m                                                ⁡                                                  (                                                                                    k                              0                                                        ⁢                            r                                                    )                                                                    ⁢                                              cos                        ⁡                                                  (                                                      m                            ⁡                                                          (                                                                                                θ                                  0                                                                -                                ϕ                                                            )                                                                                )                                                                                                                                ]                                                          (        26        )            